A frequency response function is one form of a description for the dynamics of a linear system. The dynamic relationship between two variables associated with such a system as an analytical expression is known as a transfer function. The transfer function is one way of representing the differential equations that describe a relationship between state variables or between a system input and a state variable. A linear system's transfer function can be represented as a ratio of polynomials in the complex Laplace variable s. In the field of modal analysis a technique for deriving a transfer function from an FRF is termed a “condensation algorithm.” Many known condensation algorithms attempt to find the optimal set of coefficients for the s polynomial using criteria such as least squares. Attempts to use existing known techniques in situations where the number of available measurement locations is limited have yielded unsatisfactory results. When the goal of an analysis is to automate commissioning of controllers or to quickly determine a basic model for system dynamics, a method that develops a useful result from a small number of measurements may be desirable. In some applications the ability to identify a finite number of frequencies where the transfer function dynamics have lightly damped resonances and anti-resonances is more important than the ability to identify exactly the damping factors associated with these resonances/anti-resonances. Such is the case when automating the process of tuning control loops in motion control systems. In auto-tuning applications, identification of all the significant lightly damped resonant frequencies is a requirement. The term significant indicates a resonance that has some impact on the stability or performance of a closed loop system. It is also anticipated that a multiple step approach where the real and imaginary components of each pole are identified in separate steps may lead to an overall better result. In this case heuristic rules for identifying the damped natural frequencies of poles and zeros correspond to determination of the imaginary components of the roots. Next a general constrained optimization can be performed to identify the real components by minimizing the least squared error between the original FRF and one derived for the selection of real components.
There exist a number of methods for identifying the dynamics of an oscillatory mechanical system by analyzing an FRF. Among publications that document these methods are those by Drs. Randy Allemang and David Brown. These methods either operate directly on the frequency domain FRF or operate on the impulse response time domain function obtained by applying the inverse Fourier transform to the FRF. These methods use least squares criteria to fit a large number of measurements from various measurement points on the mechanical system to determine the system eigenvalues. In a second step the residues are determined from which the transfer function matrix is derived. The same methods applied to a limited number of input-output locations have failed to provide the needed results.
In U.S. Pat. No. 6,347,255 to Moser, Moser describes a technique for identifying “poles” and “zeroes” from an FRF measurement. Moser's approach is heuristic and is designed specifically for the case of servo tuning. Software that implements Moser's approach occasionally fails to identify a pole or zero whose presence affects the tuning of the servo controller.
The current invention may also uses a heuristic approach but provides additional action and aspects to Moser's approach by imposing more restrictions on the conditions for identifying a pole/zero to gain improved immunity to noise. The result may provide a more robust method that can detect more actual poles while being less susceptible to false results, i.e. identification of a pole that is not a system pole but rather a consequence of noise in the measurement or a manifestation of non-linear effect in the system.
Accordingly, a need exists for a device, method, and system for quickly and efficiently determining the damped frequencies of a dynamic system. There may be an additional need to prevent misidentifying or identifying incorrect damped frequencies.